How do you solve quadratic inequalities and rational equations?

Cracking the Code: Quadratic Inequalities and Rational Equations Made Easy

Algebra. Just the word can send shivers down some spines, right? But honestly, it’s not as scary as it seems, especially when you break it down. Today, we’re tackling two big topics: quadratic inequalities and rational equations. Think of this as your friendly guide to conquering these mathematical beasts.

Quadratic Inequalities: It’s All About the Range

So, what’s a quadratic inequality? Forget about finding one perfect answer. Instead, we’re hunting for a range of numbers that make the inequality true. It’s like saying, “Give me all the x’s that make this thing work!”

Decoding the Standard Form

You’ll usually see these guys looking something like this: ax² + bx + c > 0 (or <, ≥, ≤ – you get the idea). The a, b, and c are just numbers, and x is our mystery variable.

How to Solve ‘Em: Two Paths to Victory

There are a couple of ways to crack these. I usually lean towards the algebraic method, but the graphical method can be super helpful for visualizing what’s going on.

Quick Example:

Let’s solve x² – 3x – 10 < 0

• Algebraically:

  • Factor: (x – 5)(x + 2) < 0
  • Critical values: x = 5, x = -2
  • After testing, you’ll find the solution is: -2 < x < 5

• Graphically: Picture the parabola. It dips below the x-axis between -2 and 5. Same answer!

Rational Equations: Taming the Fractions

Okay, rational equations. These involve fractions with polynomials, which sounds fancy, but it’s just about being careful with your algebra.

The Steps to Success

Example Time:

Solve: 1/x + 1/2 = 1/3

So, x = -6 is our answer.

Wrapping Up

Quadratic inequalities and rational equations might seem intimidating at first, but with a little practice, you can totally master them. The key is to understand the steps, work carefully, and always double-check your answers. Happy solving!